Dynamic instabilities induced by asymmetric influence: Prisoners’ dilemma game in small-world networks

A two-dimensional small-world-type network, subject to spatial prisoners' dilemma dynamics and containing an influential node defined as a special node, with a finite density of directed random links to the other nodes in the network, is numerically investigated. It is shown that the degree of cooperation does not remain at a steady state level but displays a punctuated equilibrium-type behavior manifested by the existence of sudden breakdowns of cooperation. The breakdown of cooperation is linked to an imitation of a successful selfish strategy of the influential node. It is also found that while the breakdown of cooperation occurs suddenly, its recovery requires longer time. This recovery time may, depending on the degree of steady state cooperation, either increase or decrease with an increasing number of long-range connections.